This section is entirely expository. The values of \(\zeta (s)\)  for some \(s\in \mathbb {Z}\)  and \(\zeta ^{\prime }(0)\)  will be calculated in a refreshing but non-rigorous manner. A mathematically correct derivation would require Riemann's well-known functional equation for \(\zeta (s)\) . (See e. g. Ref.[1]}) Nevertheless, it is interesting to look at these special values in the broader context of diverging sums which are interesting mathematical objects in its own right. (see e. g. Ref.[2]}).
